A ring of mass $M$ and radius $R$ is rotating with angular speed $\omega$ about a fixed vertical axis passing through its centre $O$ with two point masses each of mass $\frac{ M }{8}$ at rest at $O$. These masses can move radially outwards along two massless rods fixed on the ring as shown in the figure. At some instant the angular speed of the system is $\frac{8}{9} \omega$ and one of the masses is at a distance of $\frac{3}{5} R$ from $O$. At this instant the distance of the other mass from $O$ is

$A$ time varying force $F = 2t$ is applied on a spool rolling as shown in figure. The angular momentum of the spool at time $t$ about bottommost point is:

$A$ uniform disc is rolling on a horizontal surface. At a certain instant $B$ is the point of contact and $A$ is at height $2R$ from ground, where $R$ is radius of disc.

A uniform rod $A B$ of mass $2 \mathrm{~kg}$ and Length $30 \mathrm{~cm}$ at rest on a smooth horizontal surface. An impulse of force $0.2\  \mathrm{Ns}$ is applied to end $B.$ The time taken by the rod to turn through at right angles will be $\frac{\pi}{\mathrm{x}}\  \mathrm{s}$, where  X=____

A flywheel can rotate in order to store kinetic energy. The flywheel is a uniform disk made of a material with a density $\rho $ and tensile strength $\sigma $ (measured in Pascals), a radius $r$ , and a thickness $h$ . The flywheel is rotating at the maximum possible angular velocity so that it does not break. Which of the following expression correctly gives the maximum kinetic energy per kilogram that can be stored in the flywheel ? Assume that $\alpha $ is a dimensionless constant

Write $SI$ unit of angular momentum and dimensional formula.